9
votes
Accepted
Is the irreducible $ \mathrm{SU}(3) $ subgroup of $ \mathrm{SU}(6) $ maximal?
The Lie algebra of $SU(6)$ splits as the Lie algebra of $SU(3)$ plus a $27$-dimensional irreducible representation. (This follows from Pieri's rule: The adjoint representation is formed from the ...
8
votes
Accepted
Order of abelian subgroup of the automorphism group of an abelian group
As Nick Gill points out, one can certainly have $|B| > |A|$ if you do not assume coprimality. If you assume coprimality then yes $|B| < |A|$.
If $A_p$ is the Sylow $p$-subgroup of $A$ then $\...
7
votes
Cohomology of the partial flag variety associated with the minimal nilpotent orbit
I think this might be so standard that there is no obvious reference. The cohomology has a basis of Schubert classes. In the adjoint case, Schubert classes are indexed by the Weyl group orbit of the ...
6
votes
Advice for PhD in Algebra
This question was asked in 2020, and had a built-in five day deadline. It never received an answer but did have some helpful comments. Since this is a general question, and future users might benefit ...
Community wiki
4
votes
Accepted
Irreducible subspaces in the space of functions on Grassmannian acted by $\mathrm{GL}_n(\mathbb{F}_q)$
For $j\leq i$, functions on $\operatorname{Gr}_{j,n}(\mathbb F_q)$ map to functions on $\operatorname{Gr}_{i,n}(\mathbb F_q)$ by sending the delta function on a $j$-dimensional subspace to the ...
3
votes
Order of abelian subgroup of the automorphism group of an abelian group
Claim) Suppose $A$ is a finite non-trivial Abelian group. Let $B$ be an Abelian subgroup of ${\rm{Aut}}(A)$. Then, $|B|\leq |A|-1$ if $|A|$ and $|B|$ are coprime.
An elementary proof by induction that ...
1
vote
References for $K$-orbits in $G/B$
For what its worth the Atlas of Lie Groups and Representation software https://www.liegroups.org computes the K orbits on G/B for any (connected complex reductive) G and any (algebraic) involution. ...
1
vote
Symmetric power lift of modular forms
This can happen. For example, the dihedral group $D$ of order $10$ has two irreducible (Galois conjugate) representations $U$ and $V$ of dimension $2$ which are not twists of each other. But $\mathrm{...
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